Localization of mobile sensor nodes is indispensable for enabling Underwater Wireless Sensor Networks (UWSNs). The gathered data is not useful if it is not correlated to a specific position of the sensor node. Many applications, such as aquatic environment monitoring, target tracking , geo-routing protocols  and pollution control, require the location information. However, Global Positioning Systems (GPS) cannot be used in UWSNs because of the high attenuation of radio signal and their power hungry nature. Moreover, UWSNs are mostly based on acoustic communication systems which suffer diverse issues stemming from the aquatic conditions, such as frequency dispersion, multipath fading, limited bandwidth and energy [3, 4].
One important challenge is that underwater sensor nodes have limited resources due to the use of non-rechargeable batteries, which directly determines the network life time. Given the engineering hurdles and financial costs of battery replacement, the design of energy-efficient localization techniques becomes critical to extend the network lifetime in UWSNs. The sensor nodes’ energy is consumed mainly by packet transmission and reception, which is much larger than that of the idle listening , so adjusting the transmission power by topology control is one possible way to save energy in UWSNs. In most localization systems of UWSNs , multiple anchor nodes are required to help one sensor node to find its position. The performance of these localization methods depend on different factors such as the nodes’ initial reference position, number of sensor nodes, number of anchor nodes, ranging technique as well as the position of anchor nodes.
These facts motivate us to seek for energy-efficient solutions enabling each sensor node to find the required number of anchor nodes in view of localization by topology control. Topology control in this paper represents the process of controlling the amount/quality of neighboring nodes by transmission power control, where anchor nodes assist sensor nodes to find their locations in UWSNs. So far, only a limited number of schemes have been proposed for the localization service in UWSNs , , , especially for localization in sparse UWSNs by using topology control . However, the proposed scheme in  only considers energy-saving for anchor nodes, although the sensor nodes deployed underwater consume the bulk of the energy, causing the network lifetime to decrease prematurely.
In this paper, we leverage the benefits of topology control to achieve a high localization performance and a low energy consumption in UWSNs. In such systems, in order to successfully localize a single sensor node, multiple anchor nodes are required. This motivates us to model this problem as a Single-Leader-Multi-Follower Stackelberg game, for which we define new utility metrics for trading-off localization ability and energy efficiency with no need for new equipment, nor extra cost, unlike the existing state-of-the-art schemes in  and . In summary, the contributions of this paper are given as follows,
The considered localization problem is analyzed as a Single-Leader-Multi-Follower Stackelberg game whereby the trade-off between localization ability and energy consumption are considered in the utility functions. Optimal transmit powers for sensor nodes and anchor nodes are derived, and are shown to achieve Nash Equilibrium.
Based on our analysis, we propose the EELA (Energy-Efficient Localization Algorithm) that builds the strategic interactions among each sensor node and multiple anchor nodes for enabling energy-efficient localization.
Extensive numerical evaluations show that the proposed EELA scheme achieves about energy reduction for all nodes on average compared to the state-of-the-art approach, while achieving high quality localization coverage under reasonable error and delay levels.
The rest of the paper is organized as follows. Section II discusses the related works. Section III introduces the system model. The detailed description of the proposed EELA scheme is presented in Section IV. Simulation results and performance evaluation are shown in Section V. Finally, Section VI presents the conclusion and future work.
Ii Related Work
Ii-a Localization in UWSNs
A set of localization techniques has been proposed for UWSNs in recent years. A detailed survey about these works was given in  and . One localization scheme was presented in , which worked well in high latency networks and improved the energy efficiency as well. One-way and two-way MAC-layer message delivery were combined in this paper. However, it is only suitable for static UWSNs due to its assumption of constant propagation delays among sensor nodes. In 
, three-Dimensional Underwater Localization (3DUL) described a distributed, iterative and dynamic solution to the localization problem in the underwater acoustic sensor network with only three anchor nodes at the surface of the water. Trilateration algorithm was used to estimate the sensor node location. However, the error accumulated with the iteration increase, leading to inaccurate positioning of later sensor nodes that require the location information of new anchor nodes generated by sensor nodes in 3DUL. One localization scheme was proposed in a hierarchical underwater sensor networks consisting of surface buoys, anchor nodes, and ordinary nodes. Sensor nodes were localized by the trilateration method. However, the node density is assumed to be high in this scheme due to the long distance acoustic communication between the anchor node and the surface buoy. In , a novel scheme was proposed for long-term maritime surveillance monitoring tasks in ocean sensor networks. Liu et al.  proposed a joint solution for localization and time synchronization in mobile underwater sensor networks. The stratification effect of underwater medium was taken into account in localization. Schemes utilizing Autonomous Underwater Vehicles (AUVs) ,  and  as beacon nodes result in additional cost to the network.
Ii-B Topology Control
The problem of topology control for WSNs has been extensively studied in recent years. The topology control scheme presented in  and  started with neighbor finding, where all nodes transmit at their maximum transmission power. Later, each node computes the minimum transmission power required to maintain network connectivity. A game theoretic model of topology control to analyze the decentralized interactions among heterogeneous sensors was given in . The connectivity, the success rate and the power consumption were considered to achieve desirable network performances. Zhu et al. 
took into account the signal to interference plus noise ratio and power efficiency to solve the distributed power control issues in cognitive wireless sensor network with imperfect information. A game-theoretic power control mechanism based on the Hidden Markov Model (HMM) was employed to maximize the network lifetime.
Although there are increasing interests in UWSNs in the past several years, few works have been investigated on approaches for topology control in UWSNs, especially for the localization by using topology control. In 
, a distributed radius determination algorithm is designed for the mobility-based topology control problem. However, the energy consumption of message reception was not considered in this paper. A scale-free network model for calculating edge probability was employed to generate the initial topology randomly by Liu et al.. In order to ensure the connectivity and coverage of the network, two kinds of cluster-heads were constructed by using a topology control strategy based on complex network theory. A Single-Leader-Multi-Follower Stackelberg game, called Opportunistic Localization by Topology Control (OLTC) scheme, was proposed in  to build a localization model with high coverage and less energy consumption in sparse UWSNs. Trilateration algorithm was employed to localize the sensor node. However, in this paper, each sensor node always uses the maximum transmission power to broadcast the ‘Request’ message, which results in more energy consumption. Besides, only the energy-saving for anchor nodes is considered in OLTC. However, in many scenarios of UWSNs, energy-saving for sensor nodes is much more important, since they have a limited battery and are deployed underwater, hence directly affecting the lifetime of the whole network.
Iii System Model
Iii-a System Overview
The proposed EELA is implemented in the 3D UWSN, where is the set of anchor nodes deployed on the surface of water and denotes the set of sensor nodes deployed underwater. All nodes move passively given the water wave and underwater currents. Fig. 1 depicts the deployment scenario of EELA, where the cylinders represent anchor nodes on the water surface while dotted circles express sensor nodes which are randomly positioned underwater. Each sensor node has multiple neighboring anchor nodes within the current transmission range under power , which is expressed by . Both sensor nodes and anchor nodes can change their transmission power to maximize their own benefits. The transmission power value for each sensor node is within , corresponding to different transmission ranges within (see Eq. (1)). The transmission power value for each anchor node is within , corresponding to different transmission ranges within (see Eq. (1)). We set .
All nodes are time-synchronized.
Sensor nodes are randomly deployed underwater while anchor nodes randomly float on the water surface.
Sensor nodes are aware of their depth.
|The set of sensor nodes|
|The set of anchor nodes|
|The set of neighboring sensor nodes of th sensor node|
|The set of neighboring anchor nodes of th anchor node|
|Additional number of anchor nodes required by th sensor node|
|Total remaining energy per node|
|Transmission energy cost per unit power of th anchor node|
|Transmission energy cost per unit power of th sensor node|
|Transmission power of th sensor node|
|Transmission power of th anchor node|
|Maximum transmission range of sensor node|
|Maximum transmission range of anchor node|
|Localization ability of th anchor node|
|Ability of sensor node to find anchor nodes|
Iii-B Propagation Model
In general, acoustic communications are used in underwater environment due to the small attenuation of acoustic signals . The attenuation in an underwater acoustic channel for a signal with frequency over a distance is given as
where is a normalization constant; is the distance in meters between the sender and the receiver; is the frequency; is the absorption coefficient in dB/m. The spreading factor is an expression of the geometry of propagation where typically . Eq. (3) describes the absorption coefficient with values in dB/km ,
where is the Thorp‘s approximation for absorption loss in dB/km, and is center frequency in kHz.
Different communication ranges correspond to different bandwidths. For example, if the distance range is 10km to 100km, the bandwidth is limited to few kHz. However, 10 kHz matches the short range (from 1km to 10km) and a few hundred kHz bandwidth is available for ranges below 100m .
Iv Problem Formulation and Solution
Iv-a Proposed Single Leader Multi Follower Stackelberg Game Formulation
In Stackelberg game , two types of players (leader and followers) are used to model the hierarchy of actions. The leader moves first and selects a strategy. Based on the action of the leader, the followers choose best response strategies that maximize their utilities. Then, the leader selects one strategy to maximize its utility based on the strategies of followers. In a distributed localization scenario, a sensor node can localize itself after receiving enough location beacons from multiple anchor nodes. However, due to the random and sparse node deployment, and the mobility of nodes, a sensor node may not find enough neighbor anchor nodes. A Single-Leader-Multi-Follower Stackelberg game  is employed, where the sensor node acts as the single leader while anchor nodes are multiple followers. The sensor node acts first and chooses a transmission power to send a request message, which is similar to the leader releasing a price in Stackelberg game. Each anchor node reacts, i.e, it selects a transmission power to send reply message, after the action of the sensor node.
Iv-B Utility Function of Anchor Nodes
Anchor nodes are followers. They decide their strategies to handle the maximum number of requests from sensor nodes with minimum energy consumption.
Let be the additional number of anchor nodes required by sensor node for localization, which is defined by,
where represents the number of anchor nodes required for one sensor node to get its location and is the set of the anchor nodes in the communication range of sensor node .
The localization ability of anchor node is composed of several terms expressing different effects,
In (IV-B), the first two terms and are similar to those in the utility function in . is the ‘ability of th anchor node to resolve sensor requests’, where is the number of requests that can be handled by anchor node with transmission power and is the total number of request messages received from sensor nodes. From Proposition 1 below, we can see that of a follower (anchor node) is non-decreasing with the increase of the transmission power . The second term is the ‘ability of th anchor node to serve additional demands’. It means that only requests can be served among the total sum demand for additional anchor nodes from sensor nodes. Finally, the third term expresses the relation between the sum-transmit power received from sensor nodes and the transmission power of the anchor node . If the transmission power received from sensor nodes increases, anchor node has to handle more sensor nodes. Therefore, the localization ability decreases.
For each anchor node , the number of neighboring sensor nodes is higher or at least equal with the increase of transmission power .
Let us assume that there are sensor nodes uniformly deployed in the area with size . can be calculated by,
where is the density of sensor nodes and is the volume of th anchor node with the transmission range .
The inverse function exists and is monotonically strictly increasing with the transmission power , because . Therefore, can be represented by Eq. (8). Then, the first order partial derivative of is given in Eq. (9),
Hence, , which proves Proposition 1. ∎
Next, we define the payoff function of any anchor node by considering various factors such as energy cost, the ability to localize sensor nodes and the transmission power of sensor nodes as well as anchor nodes. It is hence defined as the weighted sum of the remaining energy ratio after transmission with and its localization ability ,
In the first term of Eq. (IV-B), is the total energy of the th anchor node and is the transmission energy cost per unit power. Weights and define the trade-off between the energy consumption of anchor node and the localization ability of anchor node, and satisfy , .
Iv-C Utility Function of Sensor Nodes
In the considered localization problem, sensor nodes act as leaders. They watch for the decision of anchors which act as followers, and based on the response of followers, maximize their profits. The strategy of the leader is to minimize the energy consumption and to localize the maximum number of sensor nodes during the allowed localization delay, which is defined in Section V-C. A sensor node broadcasts a ‘Request’ message to explore the maximum number of anchors. After sensor nodes receive enough beacon locations from neighbor anchor nodes, it will localize itself.
The ‘ability of sensor node to find anchor nodes’ is composed of two terms,
In Eq. (IV-C), the first term is the ratio of the number of anchor nodes within ‘one-hop’ and ‘two-hop’ ranges of transmission power and the total number of anchor nodes received, where is a non-decreasing function of , the proof of which is similar to that of Proposition 1. The second term is the ratio of the sum-transmission power of received anchor nodes and the transmission power of sensor node . If the transmission power of anchor nodes increases, more anchor nodes’ beacon messages are received so less anchor nodes can be reached with a given power . Therefore, the ‘ability of sensor node to find anchor nodes’ is inversely proportional to this ratio.
The payoff of any sensor node increases with the decrease in energy consumption. Also, it increases with the increase of the number of neighbor anchor nodes. In addition, the payoff of the leader decreases with each retry it does to send the ‘Request’ message. Therefore, the payoff function of sensor node is defined as Eq. (IV-C), which is the weighted sum of its remaining energy ratio and the ‘ability of sensor node to find anchor nodes’,
In the first term of Eq. (IV-C), is the total energy in the th sensor node and is the transmission energy cost per unit power. Weights and provide a trade-off between energy consumption and ‘ability to find anchor nodes’, satisfying , .
Iv-D Existence and Uniqueness of Stackelberg Nash Equilibrium
The considered game achieves equilibrium when the sensor node (leader) selects the optimal transmission power to get its location with minimum energy consumption while anchor nodes (followers) choose their optimal transmission power to localize the maximum number of sensor nodes with minimum energy cost. At equilibrium, the benefit of each side can not be improved by unilaterally changing its own strategy. To find the Stackelberg equilibrium, each sensor node calculates the best reaction of anchor nodes to each of its mixed strategy and selects the mixed strategy that maximizes its own utility.
Iv-D1 Best Response Strategy of Anchor Nodes
To define the strategy of the th anchor node , the transmission power allocation problem can be cast as the optimization problem formulated below,
All anchor nodes are non-cooperative. In Proposition 2, the existence of the best response strategy of each anchor node is proved and the unique equilibrium point is computed.
Let be the strategy of the th anchor node. The best response of each anchor node is given as,
with , and , where
We prove that problem (13) is a standard form of convex optimization and determine the expression of the optimum.
The first order partial derivative of with respect to , for , is given as
Let , then we get as,
The second order partial derivative of is given as
Here, we need to prove in order to prove that is negative, where
The inverse function of anchor node exists and is strictly monotonically increasing with the transmission power , because and . Then, can be calculated as
Then, we have
from which we get for (IV-D1) .
where and are given as,
From Eq. (3), we know , from which we get , where and . Therefore, is proved.
Since the value of the second order partial derivative of is negative, the maximum value of can be achieved at by solving , proving Proposition 2. ∎
Iv-D2 Best Response Strategy of Sensor Node
To define the strategy of the th sensor node , the transmission power allocation problem can be cast as the optimization problem formulated below,
The existence and uniqueness of the sensor node’s transmission power at Nash equilibrium is proved in Proposition 3.
Let be the strategy of the th sensor node. The best response of each sensor node is given as,
with , and , where , and
We prove that problem (23) is a standard form of convex optimization and determine the expression of the optimum.
The first order partial derivative of with respect to is given as
Letting , we get
where and .
To guarantee the existence of , we need to set the condition .